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Motion of Electron in Ionized
Gasses : Monte Carlo Simulation of Huxley-Townsend Experiment
- Laureando : Stefano Pasquali Relatori : Chiar.mo Prof.
Gianlorenzo Braglia,
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Chiar.mo Prof. Giuseppe Mambriani
 | Cross Section Calculation |
| Facing The Idrogen Problem |
| Comparison between Theory and Experiments |
| Monte Carlo Simulation |
| Source Codes Ottimization |
| Comparison MC-Teory, MC.Experiments |
| Comparison between different Monte Carlo Simulations. |
Introduction
For nearly
thirty years in literature there has been a very interesting problem called "H2 problem"
which has not been satisfactorily answered yet. It is the discrepancy between the
vibrational theoretical cross sections and those experimental-ones obtained by means of
commonly considered very accurate techniques. The case e - H2 is the most
simple system of scattering electron-molecule and so it has been taken into considerable
attention both theoretically and experimentally.There is a satisfactory arrangement
between moment transferring and rotational cross sections obtained by " swarms "
technique and the theoretic prediction. On the contrary, there is a significant
disagreement concerning vibrational cross sections (this discrepancy is not present, on
the contrary, in " beams " electronic experiments.
In this graduation thesis we:
1. discuss the whole
transport theory to show the validity of some approximations introduced in the
determination technique of cross sections by experiments in Townsend-Huxley.
- -Development in two terms of the electronic distribution function of
Boltzmann equation.
- -Reliability of the rigid spheres model in the scattering treatment.
- -Hypothesis on the isotropic scattering .
Then resorting to Monte Carlo simulations:
2. You discuss again the
whole previously developed Monte Carlo treatment which had perplexed a lot on the
"swarms " technique with modernized or totally rewritten codes, where you
discuss theoretically and with computation the iniectivity of simulations with the
physical reality; you do exausting tests on the numeric processes and on the introduced
approximations.
3. Besides you run through
the whole old series of simulations again with much more elevated statistics to see the
probable dependence of the results on "bugs" of the same Monte Carlo. Moreover
the comparisons between the new MC data transport coefficient, drift velocity and
current ratios.
4. After removing even
doubt on used algoritmi reliability (3), we passed to "gas models". So we could
compare different simulations to investigate in a qualitative way, on the possible reasons
of such discrepancies. In particular one thinks the problem should be in the conditions
and hypotheses situated in the Diffusion Theory used, in Townsend-Huxley experiment, to
infer the cross sections.
Among the numerous executed comparisons one is particularly
interesting. Using a new code, we managed to make a differential cross section ( as a
proof ) which let us treat the e-H2 scattering as anisotropic with different
distributions of cos c ( where cos c is the scattering angle ). The discrepancy
between the isotropic case ( considered in the " swarms " technique ) and the
anisotropic case is, at the most, about 3% on current ratios end 4,5% on Dt /
m in gas model and it depends
on the range of the considered energies. We think this is the first real indication on the
accuracy of one of the essential hypotheses situated on physical system (isotropc
scattering).In fact the expected entity of the discrepancy, such as to allow the solution
of "hydrogen problem", is about 7-8 % on Dt / m in real gas. As the second result of the
great work done according to computation, we think we have now at our disposed a Monte
Carlo code which is scrupulous and has elevated statistics. It also applies to scattering
problems different from that one simulated. For example, an emerged realistic application
concerns the ottimization of radiotherapeutic techniques for the selective elimination of
cancerous cells.
The transport theory of electrons in gas
The conventional theory
- -The number of electrons in comparison with
the number of gas molecules is negligible.
- -The electron - electron crosses have
negligible effects. So we have the scattering problem.
-
- The definition of "swam".
- -The electrons come into collision with the
molecules with binary crosses ( the gas is diluted ).
- -The particles which make the gas are
casually distributed during the phases
- (absence of correlation Molecular
Caos )
Boltzmann Equation
Hypothesis 1: Two terms approximation.
Hypothesis 2: Absence of correlation between r
and v
Integral and frequency of collision
The transport equation for f and n
The transport parameters
Limits and overcoming of the conventional
theory
- -For the conventional theory Dl =
Dt = D ( Isotropic scattering ).
- -(1967). One finds out experimentally the
scattering is anisotropic.
Monte Carlo simulations confirm the approach
and the limits of the conventional theory.
There is a correlation Position - Velocity
- The distribution described by Boltzmann equation coincides
only asymptotically with that
- one produced by Monte Carlo for T elevated enough.
- Continuity
Monte
- Equation
Carlo
The cross sections determination
| Theoretic determination from first principles. (not developed
here) |
| Experimental methods. (in the following we'll introduce one of
this methods) |
Huxley - Townsend experiments
Description of experimental apparatus.
- The electrons, released for photoelettric effect by the
metallic plate F shunt and diffuse towards the A anode, in an electric field kept uniform
in the cylindrical ambient by the electrodes of guard G1.........G5.
The C cathode has a central hole with a radius a which is the real and right source of
electrons.One assumes that the initial conditions to the cathode are well defined, that
is:
-
- 1. The electrons
enter the scattering volume with a distribution in velocity formerly stationary, which is
the same as that - one which is between C and A.
- 2. The distribution
of density n at the source is known and one can assume it null ( n = 0 ) on the cathode
for all the r > a.
-
- We notice that the condition -1- can be satisfied if you let
the electronic cloud shunt for an adequate distance, before reaching the cathode. Besides,
as regards the source, because it has finite dimensions (a ~ some mm.), it is used to
assume that the density of electrons is constant everywhere in the opening. The anode
consists of two concentric electrodes. One interior with radius b and the other with
radius c.
- Huxley - Towsend experiment consists in measuring the currents
ib and ic, obtained for a given value d to the electrodes of the
anode in order to manage to calculate the ratio.
- From this measures it is then possible to get Dt
through some appropriate relations we are going to introduce.
- We know W ( previously fixed ) and E (applied electric
field).In order to get the relation R = R ( Dt).
- A lot of analytical processes have been elaborated. Now we
shortly show those principal - ones we refer to.
-
1.
Townsend formulation.
2.
Huxley formulation.
3.
Lowke formulation
Method for cross sections determination
1) Transport parameters Measure
2a), 2b) "Ad hoc " cross
sections construction.
3) Computation of transport
coefficients from Boltzmann definite equation.
4) Comparison of the coefficients
sets and cross sections " tuning " until there is a " matching ".
Problem of the cross sections unicity
( uncertainty > 10% )
Monte Carlo Method
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- Probability Density Function
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- Pseudo casual numbers generator ( in
the foundamental version it generates numbers uniformly distributed in [ 0,1 ] ).
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- Samplings rules, with the random numbers of PDF.
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- Mistake valutation according to the extractions number.
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- Methods for the statistical fluctuation reduction
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Monte Carlo artifices
| Null event technique. |
The flight and the extraction technique of the collision
frequency scheme (Q ).
| Distribution just before the cross. |
| Definition of the sources. |
Tests on Random Number Generator
("Best Seed" (M0) for
random extraction determination).
| Spectral test relative to 106 extractions with Seed
>> M0 ( << M0 ). |
| Spectral test relative to 107 extractions with Seed
@ M0. |
Executed simulations and relative
comparisons (in steps 1,2,3)
1. Already existing
in real gas ( H2 ) Monte Carlo measures refining.
Current Ratio for Normal Idrogen. E/N = 15,17 Td ;
T=77°K. Diffusion Room Lenght 10 cm. Source Diameter 1 mm
According to all these new results, which have to cheek those
- ones previously obtained with other codes and lower statistics and with less powerful
computers, we can state that:
- Huxley - Townsend experiment seems to result
autoconsistente ; the approximations which are present in its theoretical-mathematical
description do not affect very much because the simulation gives the experimental values,
within the mistake limits.
-The discrepancy found in the hydrogen vibrational cross
sections is not due to the accuracy lack of the results pertinent to this experiment ( Dt / m ) experimental conditions of interest.
In order to confirm what above declared in the end we aded
that the experimental value of Dt /
m (that is
obtained from R by the theoretic formulas) has been obtained with a precision inferior to
1% so consistent with the experimental mistake, through a direct transport parameters
simulation in infinite gas (that is in absence of absorbing walls).
- 2. Search of possible
discrepancies among simulations which neglect or not the recoil effect in the cross (in
model gas). In these cases one noticed the per cent difference on R values, for E/N values
we are interested in, is about 0,18%. In conclusion the discrepancy in the current
connection R between Monte Carlo and experimental data is not attributable to possible
placed approximations which neglect the recoil.
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- 3. Simulations comparisons with
isotropic and anisotropic scattering ( in model gas H2 ). In this kind of
simulations we concentrated our attention on model gas " Simil H2 ".
We made simulations according to the experimental data Sets and the used scattering type.
So we managed to investigate on the different approximations weight on the scattering (
isotropic, anisotropic 90°, anisotropic 180° ) and so to try to confirm or not the third
hypothesis supposed by the theoretical analysis of the Huxley - Townsend experiment
data. For every kind of simulation, identified by the experimental conditions and the
scattering type, we calculated 17 R values, whose average was used for the comparisons
according to the operative diagram said at the end of the fifth chapter. From the tests
done on Monte Carlo we established further simulation parameters to assure an elevated
fluctuations. The relative characteristics of every single simulation ( among the total
255 ) are the following:
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- -The seed of the random generator chosen according to the
previous indications.
- -The whole number of events ( null or real ) @ 1012. Fixed according to the life
simulation constant.
- -The whole number of mill crosses @ 1011. Dependent on the epsgam
simulation constant which says, among the whole crosses, how many of them are null.
- -So that the electronic cloud can be considered in an almost
stationary state ( absolutely necessary condition ) the energetic balance B ratio between
energy given from the field and energy lost in the numerous crosses) must result 1 (in
every simulation we developed (1 - B) < 10 -9 ).
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- From a first analysis of the simulation R values, two
predominant characteristics which have a physical reply emerge:
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- R anisotropic 90° results lower than the isotropic one
because the electrons spread more and so they are more absorbed by the external electrode.
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- R anisotropic 180° is higher than the relative R isotropic.
This is due to the fact that deflections 90° are really less probable and so the
electrons are not used to giving up, for lateral scattering,the cylindrical symmetry axis
passing through the source centre, reaching the internal electrode in the great number.
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- In order to value the importance or not of the anisotropic
hypothesis, we compute the per cent R variation from the isotropic case for every kind of
executed simulation. We define the following quantities:
A graphic representation of DR 90 and DR 180 trend in
function of E/N is show in the following figure. You can well notice the corrections given
to R value in the two different cases. Even if you are working in model gas, such a result
is very significant because for an anisotropic scattering 180° the corrections go to the
right direction and let us think that it is so possible to get a good arrangement between
the experimental data and Monte Carlo simulation.
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